Macroscopic Orientation of Inertial Flows in Porous Media
Bibliographic record
Abstract
Abstract Macroscopic models of inertial flows in porous media have many practical applications where direct numerical simulations are not feasible. The Forchheimer equation describes macroscopic momentum transport accounting for inertial effects at the pore scale through a nonlinear correction tensor $${\textbf{F}}_\beta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>β</mml:mi> </mml:msub> </mml:math> to the permeability. The goal of this work is to study the effects of inertial flow orientation on the Forchheimer correction. Using up-scaling approaches such as the volume averaging method, $${\textbf{F}}_\beta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>β</mml:mi> </mml:msub> </mml:math> can be determined. However, the procedure requires to deal with a nonlinear problem for the deviations of the local velocity field. This is commonly tackled by assuming that the inertial convective velocity is decoupled from the velocity deviations. Here, we propose an alternative approach based on regular perturbation expansion leading to a series of linear closure problems. The values of $${\textbf{F}}_\beta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>β</mml:mi> </mml:msub> </mml:math> predicted by both approaches are compared for various values of the Reynolds number and flow orientation. Compared to the local inertial–convection approach, the proposed linearized closure problem has the advantage of being self-consistent, independent of the pore Reynolds number and of flow orientation. It is, however, limited in validity by Reynolds number below one and requires the solution of closure problems of higher dimensions. Then, macroscopic simulations are performed to evaluate the importance of varying pressure gradient orientation on the macroscopic inertial flow. Numerical results of the general macroscopic model obtained by the volume averaging method highlight the necessity to account for extra-diagonal terms as well as macroscopic gradient orientation in the determination of the Forchheimer tensor.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".